HYPOTHESES OF DYNAMICS. 5 



To find this iu the case of the first law we need ouly note that as according to this 

 law two particles which are both free from the action of force must have uniform velocities 

 relatively to the unspecified point and axes of reference, each must have a xmiform 

 velocity relatively to the other and to axes through it parallel to the unspecified axes, 

 and therefore also relatively to the other and to any axes through it which have a con- 

 stant inclination to the unspecified axes. Now it follows from the first law also that the 

 lines joining particles which are free from the action of force and have the same A'elocity 

 must have constant inclinations to the unspecified axes. Hence the first law holds 

 relatively to any particle, not acted upon by force, as point of reference, and to lines drawn 

 from it to other particles which are unacted on by force and have the same velocity as the 

 first particle, as axes of reference. We may say for shortness that it holds relatively to a 

 set of particles, free from the action of force and having the same velocity, it being under- 

 stood that the set is at least four in number and that they are not co-planar. It may 

 therefore be enunciated so as to be quite precise, in the following words : — Relatively to 

 a set of particles free from the action of force and having the same velocity, the velocity 

 of a body which is not acted upon by force is uniform. — It may be noted that it is always 

 .possible to select from a group of particles on which no forces act, those which have the 

 same velocities, by taking any one of the group as point of reference, and lines drawn from 

 it to other three, as axes of reference. Though, relatively to these axes, the velocities of 

 the other particles will in general be variable, we may by using these axes select the 

 particles whose velocities at any instant are the same. 



Thomson and Tait regard the laws of motion as holding relatively to the fictitious 

 fixed point and axes, and show iu the following passage,' how we may imagine our- 

 selves as obtaining directions of reference fixed relatively to the fixed axes : — " If two 

 material points be projected from one position. A, at the same instant, with any velocities 

 in any directions, and each left to move uninfluenced by force, the line joining them 



will be always parallel to a fixed direction Hence, if four material points 



O, P, Q, R, are all projected at one instant from one position, OP, OQ, OR, are fixed 

 directions of reference ever after." It will be obvious that this mode of obtaining a 

 point and axes by reference to which the first law holds, is quite independent of the as- 

 sumed fixedness of Newton's unspecified axes, and that it is essentially the same as the 

 mode employed above. 



Tait regards the first law as holding " relatively to any set of lines drawn in a rigid 

 body of finite dimensions, which is not acted on by force and which has no rotation." ' 

 This mode of specifying the relativity of the first law is clearly the same as that employed 

 above. But it is open to the objection that the first law by itself gives us no means of 

 recognizing a rigid body as not rotating ; and that unnecessary reference is thus made to 

 the other laws of motion. 



The second law of motion states that the acceleration of a body is in the same direction 

 as the force which produces it, and is proportional to the quotient of the force by the 

 mass of the body. To determine a point and axes by reference to which it holds, it is only 

 necessary to find a point and axes by reference to which the acceleration will have the 



1 Treatise on Natural Philosophy, Vol I, Part 1 (1879), ? 249. 



2 Properties of Matter (1885), p. 92. 



